For DL WPT, the HAP broadcasts an energy signal with constant transmit power \(P_{\rm A}\)  for a duration of \(\tau _0\) .
The energy harvested from the noise is assumed to be negligible as in [1]}, since the noise power is much smaller than the power received from the HAP. Let \(\mathbf {\Theta }_0 = \text{diag} ( e^{\jmath \theta _{0,1}}, \cdots , e^{\jmath \theta _{0,N}})\)  denote the reflection phase-shift matrix of the IRS for DL WPT where \(\theta _n\in [0, 2\pi ), \forall n\) .
Thus, the amount of harvested energy at device \(k\)  can be expressed as
\(E^h_k=\eta _kP_{\rm A}|h^H_{d,k} + \mathbf {h}^H_{r,k}\mathbf {\Theta }_0 \mathbf {g}|^2\tau _0 =\eta _kP_{\rm A}|h^H_{d,k} + \mathbf {q}_k^H \mathbf {v}_0|^2\tau _0,\) 
